Multi-Bubble Blow-up Analysis for an Almost Critical Problem

Consider a smooth, bounded domain $\text{Ø}\subset {\mathbb{R}}^n$ with $n\ge 4$ and a smooth positive function $V$. We analyze the asymptotic behavior of a sequence of positive solutions $u_{\text{e}}$ to the equation $-\mathrm{\Delta }u+V(x)u=u^{\frac{n+2}{n-2}-\text{e}}$ in $\text{Ø}$ with zero Dirichlet boundary conditions, as $\text{e}\to 0$. We determine the precise blow-up rate and characterize the locations of interior concentration points in the general case of multiple blow-up, providing an exhaustive description of interior blow-up phenomena of this equation. Our result is established through a delicate analysis of the gradient of the corresponding Euler-Lagrange functional.